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Polygons, Polyhedra, and Polytopes Polygons, Polyhedra, and Polytopes Marco Gualtieri Department of Mathematics, University of Toronto Warm-up: Polygons A polygon is a region of the plane whose border is a path made up of straight line segments which only touch at endpoints (called vertices). We say the polygon is convex when it \bulges out": a line joining two points in the polygon never leaves the polygon. Regular Polygons We say that a convex polygon is regular when all its sides have the same length and all the angles are the same. f3g f4g f5g f6g f7g f8g We can focus on the interior of the polygon, which is 2-dimensional, or on its boundary, which has dimension 1. Regular Polygons: Calculation To compute the angles, drive around the polygon { you will turn n times, each time by 1=n of a full turn. For a pentagon, the interior angle is 3=10 of a full turn, i.e. 108◦. Polyhedra A Polyhedron is a region of 3D space with boundary made entirely of polygons (called the faces), which may touch only by sharing an entire edge. We want only convex polyhedra: a line joining two points in the polyhedron must be completely contained in it. Also we want regular polyhedra: this means that the faces are all the same regular polygon and the vertices are all identical. Regular Polyhedra If the faces are p-gons, and there are q around a vertex, then 1 1 1 1 each contributes an angle 2 − p , for a total of q( 2 − p ). But this must be less than a full turn (or the vertex would be flat), so: 1 1 1 + > q p 2 Faces used Solution Schl¨aflisymbol Triangles (p = 3) q = 3, 4, 5 f3,3g, f3,4g,f3,5g Squares (p = 4) q = 3 f4,3g Pentagons (p = 5) q = 3 f5,3g f3,3g: Tetrahedron If we assemble three regular triangles we see that the free edges form another regular triangle, so we can glue in a fourth triangle to get a f3; 3g. This is the Tetrahedron. I Show that non-intersecting edges are perpendicular. I Show that the angle between two faces is cos−1(1=3) ≈ 70:5◦. f3,4g: Octahedron The height of a pyramid on a square can be adjusted until the triangles are regular. Then two of these placed base to base gives a polyhedron where every vertex has four regular triangles: it's a f3; 4g. This is the Octahedron. In Cartesian coordinates the vertices are (±1; 0; 0); (0; ±1; 0); (0; 0; ±1): Looking at the following picture, we see that the dihedral angle of the octahedron and tetrahedron sum to a 1=2 turn, giving ≈ 109:47◦ for octahedron. f3,5g: Icosahedron By adjusting the height of a pyramid on a pentagon, we get five regular triangles to a single vertex. Then observe that two of these can be glued on an antiprism: This forms a f3; 5g: the Icosahedron (twenty \seats" or faces). It has dihedral angle ≈ 138:19◦ f4,3g: Cube Finally we get to the cube, which is just a prism on a square: each vertex has three squares. Dihedral angle is 90◦. The cube may also be constructed as the dual of the octahedron. It may also be synthesized as a hydrocarbon C8H8 called cubane. f5,3g: Dodecahedron To build f5; 3g, assemble a \bowl" of six pentagons and show that the free vertices have the same distance from the central axis. This shows that two such bowls may be glued together. Alternatively, it is the dual to the icosahedron. The vertices have the following coordinates: (±1; ±1; ±1); (0; ±φ−1; ±φ); (±φ−1; ±φ, 0); (±φ, 0; ±φ−1); p 1+ 5 where φ = 2 is the golden ratio. The dihedral angle is 2 tan−1(φ) ≈ 116:57◦. Of course, it has also recently been synthesized as Dodecahedrane C20H20. Polyhedra Summary Regular Polytopes An n-polytope is region in n-dimensional space with boundary made entirely of n − 1-polytopes (called the faces), which may touch only by sharing an entire one of their own faces. We focus on convex polytopes which are also regular, which means they have the greatest possible symmetry: any way of picking a face, and then a face of that face, continuing down to a point, is equivalent to any other. A note about visualization: When imagining a 4-d polytope, it is useful to concentrate on its 3-d boundary: this is a bunch of 3-d polyhedra which are glued together along their 2-d faces. Recall that we sometimes describe 3-d polyhedra by describing their 2-d surface. The most obvious case of this is its net, a collection of planar polygons which folds to give the surface of the polyhedron. The net of a polytope is a collection of polyhedra glued along their faces: the result is the 3-d boundary of the 4-d polytope. Another strategy we use for polyhedra is to project the boundary surface to 2-d with transparent faces. We obtain a diagram of overlapping 2-d polygons which show how the polygons on the boundary are glued together. On the left, we understand how two of the overlapping squares are glued along an edge to make part of the boundary of a 3-d cube. On the right, we see how three of the overlapping cubes make part of the boundary of a 4-d cube. Note that this is possible because the dihedral angle of the cube is 90◦, so three of them come together to make a convex edge. Possible 4-polytopes The constraint that the dihedral angle must sum to less than a full turn at an edge classifies polytopes in dimension 4. Faces used Dihed. angle # at each edge Schl¨afli Tetra. (f3,3g) 70:53◦ 3,4,5 f3,3,3g, f3,3,4g,f3,3,5g Octa. (f3,4g) 109:47◦ 3 f3,4,3g Icos. (f3,5g) 138:19◦ none Cubes (f4,3g) 90◦ 3 f4,3,3g Dodec. (f5,3g) 116:57◦ 3 f5,3,3g All 6 of these possibilities are realized by an actual polytope, and these were first classified by Schl¨afli. f3,3,3g, f3,3,4g, and f4,3,3g These three polytopes exist in all dimensions: they are the Simplex, the Cross polytope and the Hypercube. The n-simplex is a pyramid over the n − 1 simplex, the n-cross is a double pyramid over the (n − 1)-cross, and the n-cube is a prism over the (n − 1)-cube. f3,3,5g: The 600-cell There is only one way of combining tetrahedra, five to an edge. The result involves 600 tetrahedra, 1200 triangular faces, 720 edges and 120 vertices. f3,4,3g: The 24-cell f3; 4; 3g is a 4-polytope whose boundary is composed of 24 octahedra joined 3 to an edge. There are 96 edges, 96 triangles, and 24 vertices (it is self-dual, like the simplex). f5,3,3g: The 120-cell The last exotic regular 4-polytope is the 120-cell, whose boundary has 120 dodecahedra joined 3 to an edge. It is dual to the 600-cell. (for the regular polytopes and the slides!) For more information, consult I Regular Polytopes, by H. S. M. Coxeter. I Geometric Folding Algorithms, by Demaine and O'Rourke. I Wikipedia's extensive Polytope data..
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